Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: The higher-point functions

We consider the critical spread-out contact process in Z^d with d ≥ 1, whose infection range is denoted by L ≥ 1. In this paper, we investigate the higher-point functions τ^(r)_ t^^→ (x^^→) for r ≥ 3, where τ^(r)_ t^^→ (x^^→) is the probability that, for all i = 1, . . . , r − 1, the individual located at x_i ∈ Z^d is infected at time t_i by the individual at the origin o ∈ Z^d at time 0. Together with the results of the 2-point function in [16], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper critical dimension 4. We also prove partial results for d ≤ 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation in Z^d × ԑZ_+, where ԑ ∈ (0,1] is the time unit. For ordinary oriented percolation (i.e., ԑ = 1), we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in ԑ ∈ (0,1], which allows us to establish the scaling results also for the contact process (i.e., ǫ ↓ 0). We also show that the main term of the vertex factor V, which is one of the non-universal constants in the scaling limit, is 2 − ԑ (= 1 for oriented percolation, = 2 for the contact process), while the main terms of the other non-universal constants are independent of ԑ.